Atmospheric Radiative Transfer

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					           Lecture 4
Atmospheric Radiative Transfer;
   Role of clouds on climate

        GEU0136 Climatology
2-Layer Atmosphere
          Radiative Balances by Layer

       For every layer:
                 Energy In = Energy Out

  TOA                (1   p )   T1


  L1                       T2 4  2 T14

  L2              Ts   T  2 T2
                              4           4

              (1   p )   T2  2 Ts
                               4        4
 2-Layer B.B. Atmosphere (cont’d)
                                      • Solving energy
                                        budgets for all layers
                                        simultaneously gives
                                                 S0 4 1   p 
                                       TS  3                         3Te
                                         4                                   4

                                      • Recall from Lecture 3
                                        that a 1 layer B-B
                                        atmosphere produces
Vertical temperature profile for 4-     Ts4 = 2Te4
layer atmosphere, with thin           • In general, an n-layer
graybody layers at top and bottom.      B-B atmosphere will
                                        have Ts4 = (n+1)Te4
Very unrealistic lapse rate!!
Molecular Absorbers/Emitters
            • Molecules of gas in the
              atmosphere interact with
              photons of electromagnetic
            • Different kinds of molecular
              transitions can absorb/emit
              very different wavelengths
              of radiation
            • Some molecules are able to
              interact much more with
              photons than others
            • Different molecular
              structures produce
Atmospheric Absorption
                 • Triatomic
                   modelcules have
                   the most
                   absorption bands
                 • Complete
                   absorption from
                   5-8 mm (H2O)
                   and > 14 mm(CO2)
                 • Little absorption
                   between about
                   8 mm and 11 mm
Line Broadening

   • Molecular absorption takes place
     at distinct wavelengths
     (frequencies, energy levels)
   • Actual spectra feature absorption
     “bands” with broader features.
    Pressure broadening
      – Collisions among molecules dissipate
        energy as kinetic (Lorentz profile)
    Doppler broadening
      – Relative motions among molecules and
        photons (Doppler profile)
Sun-Earth Geometry




      • Radiance is energy per
        unit solid angle, usually
        referred to in a given
        band of wavelengths
      • Flux (or irradiance) is the
          total energy passing
          through a plane
          (integral of radiance)

      •   q = zenith angle
      •   j azimuth angle
      •   dw solid angle increment
         Solar Absorption
                     • Absorption
                       depends on path
                       length through
                       the atmosphere,
                       not vertical
                     • dz = ds cos q
                     • ds = dz / cos q

dF  kabs  a Fds
    Beer’s Law (absorption)
dF  kabs  a Fds

                (convert from ds to dz)

               (define optical depth)

               (optical depth is a convenient coord!)

                             Exponential “decay” of
                             radiation as it passes
                             through absorbing gas
Atmospheric Absorption and Heating
             (density of absorbing gas decreases with z
             H is scale height = RT/g)
                 (optical depth as a function of height
                 and mixing ratio of absorber)

                 (differentiate and divide … simple
                 relationship between optical depth and z)

             (Heating rate is proportional to flux divergence)

                      m  cos q
                                  Absorption      Local flux
Absorption (Heating) Rate (cont’d)
  Where is max heating?

  Find out by differentiating previous equation
  w.r.t. t, setting to zero, and solving for t

                      not 0     t/m = 1

• Maximum absorption occurs at level of unit
  optical depth
• Higher in the atmosphere as sun is closer to
Thermal Absorption and Emission
                • Upwelling terrestrial
                  radiation is absorbed
                  and emitted by each
                • As with solar radiation,
                  path length ds is the
                  distance of interest,
                  rather than dz
                • Also have to consider
                  solid angle dw
        Infrared Radiative Transfer
      For radiance of a given frequency n
      passing through a thin layer along a path ds

                         dIn  En  An
                          emission     absorption (Beer’s Law)

                                     Kirchoff’s Law: an = en
              Planck                    so en = a ds kn
                                                Planck       intensity

  Gathering terms:
 Infrared Radiative Transfer (cont’d)

    Previous result:

     Convert to z:

Define optical depth from surface up:

Rewrite result in t coordinate:
          IR Radiative Transfer
               Schwarzchild’s Equation

Previous result:

                                     tn / m
    Multiply by integrating factor   e
     Interpretation for Schwarzchild’s

Radiance at a                           Sum of      weighted by
                Emission   Absorption
given optical                           emissions   absorptivity/
                from sfc
depth (z) and              below t      from each   emissivity of each
angle                                   atm level   layer in between

 • Upwelling radiance at a given level has
   contributions from the surface and from every
   other level in between
 • Relative contributions are controlled by vertical
   profiles of temperature and absorbing gases
    Simple Form of Schwarzchild’s
 Integrate Schwarzchild across thermal IR and
 across all angles and make simplifying assumptions
 to obtain simpler expressions for upwelling and
 downwelling radiative fluxes


   blackbody emission          transmission functions
   (temperature dependence)    (emissivity and radiances)

           IR Fluxes and Heating
     Net flux(z):

     Heating rate:


  IR at sfc

OLR and downward IR at surface depend on
  temperature profile and transmission functions
  Transmission Functions and Heating
                              • Think of upwelling and
                                downwelling IR as
                                weighted averages of T4
                              • The change in
                                transmission function
                                with height is the
                                weighting function
                              • Downwelling IR at
                                surface comes from
                                lower troposphere
                              • Upwelling IR at TOA
                                comes from mid-upper
Vertical profiles of            troposphere
atmospheric LW transmission   • This is the very basis
functions and temperature       for the so-called
                                “greenhouse effect”
Cloud Radiative Properties
      Cloud Radiative Properties:
          Dependence on Liquid Water Path

• Recall a + r + t = 1
• Thick clouds reflect and absorb more than thin (duh!)
• Generally reflect more than absorb, but less true at
  low solar zenith angles
Cloud Radiative Properties:
     Dependence on Drop Size

                      • Small droplets make
                        brighter clouds
                      • Larger droplets
                        absorb more
                      • Dependence on
                        liquid water path at
                        all droplet sizes too
      Cloud Radiative Properties
                Longwave Emissivity

• Clouds are very good LW absorbers.
• Clouds with LWC > 20 g/m2 are almost blackbodies!
       Radiative-Convective Models
                (a recipe)
• Consider a 1-D atmosphere                        Tn   en
• Specify solar radiation at the
  top, emissivity of each layer
• Calculate radiative equilibrium
  temperature for each layer
• Check for static stability
• If layers are unstable, mix them!
   – (e.g. if G > Gd, set both T’s to mass-
     weighted mean of the layer pair)              T3   e3
• Add clouds and absorbing gases
                                                   T1   e1
  to taste

                     Manabe and Strickler (1964)
Radiative-Convective Equilibrium
                    • Pure radiative
                      equilibrium is way
                      too hot at surface
                    • Adjusting to Gd still
                      too steep
                    • Adjusting to
                      observed 6.5 K km-1
                      produces fairly
                      reasonable profile:
                       – Sfc temp (still hot)
                       – Tropopause (OK)
                       – Stratosphere (OK)
Radiative-Convective Equilibrium
     Effect of Different Absorbers

                             • Water vapor
                               alone …
                               atmosphere is
                             • H2O + CO2 …
                               almost 10 K
                             • H2O + CO2 + O3
                               … stratosphere
Radiative-Convective Equilibrium
        Radiative Heating Rates

                        • L indicates longwave
                        • S indicates heating (by
                          solar absorption)
                        • NET combines all
                        • Heating and cooling
                          nearly balance in
                        • Troposphere cools
                          strongly (~ 1.5 K/day)
                        • How is this cooling
                           – In the R.C.M?
                           – In the real world?
        Radiative-Convective Equilibrium
                              Effects of Clouds
                                          • Clouds absorb LW
                                          • Clouds reflect SW
                                          • Which effect “wins?”
                                          • Depends on emitting T
                                          • For low clouds, T4 ~ Ts4 ,
                                            so SW effect is greater
                                          • For high clouds, T4 << Ts4
                                            so LW effect “wins”
                                          • High clouds warm
                                          • Low clouds cool

Details are sensitive to optical
properties and distributions of
clouds, but remember the
basic conclusions
Observed Mean Cloud Fraction
        high clouds
        ( < 440 mb)
                      • High clouds mostly due to
                        tropical convection
                        (Amazon, Congo,
                        Indonesia, W. Pacific)
        low clouds
        ( > 680 mb)   • Low clouds
                        (stratocumulus) over
                        eastern parts of
                        subtropical ocean basins
                         – Cold SST
                         – Subsiding air
        all clouds
                         – Strong inversion
Annual Mean Cloud Forcing
        D OLR         • “Cloud forcing” is
                        defined as the
                        difference between a
                        “clearsky” and “all sky”
        D solar abs   • At the surface,
                        (a) is all warming, and
                        (b) is all cooling
                      • Net effect of clouds
                        is to cool the surface,
       D Rnet           but changes can go
                        either way
Global Mean Cloud Radiative Forcing

 • Clouds increase planetary albedo from 15% to 30%
 • This reduces absorbed solar by 48 W m-2
 • Reduced solar is offset by 31 W m-2 of LW warming
 • So total cloud forcing is –17 W m-2
 • Clouds cool the climate. How might this number change if
   cloudiness increased?

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